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Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: applications to contour lines, sparse data and inpainting

Zhang, Kewei; Crooks, Elaine; Orlando, Antonio

Authors

KEWEI ZHANG Kewei.Zhang@nottingham.ac.uk
Professor of Mathematical Analysis

Elaine Crooks

Antonio Orlando



Abstract

This paper is concerned with applications of the theory of approximation and interpolation based on compensated convex transforms developed in [55]. We apply our methods to (i) surface reconstruction starting from the knowledge of finitely many level sets (or ‘contour lines’); (ii) scattered data approximation; (iii) image inpainting. For (i) and (ii) our methods give interpolations. For the case of finite sets (scattered data), in particular, our approximations provide a natural triangulation and piecewise affine interpolation. Prototype examples of explicitly calculated approximations and inpainting results are presented for both finite and compact sets. We also show numerical experiments for applications of our methods to high density salt & pepper noise reduction in image processing, for image inpainting and for approximation and interpolations of continuous functions sampled on finitely many level sets and on scattered points.

Citation

Zhang, K., Crooks, E., & Orlando, A. (2018). Compensated convexity methods for approximations and interpolations of sampled functions in Euclidean spaces: applications to contour lines, sparse data and inpainting. SIAM Journal on Imaging Sciences, 11(4), 2368-2428. doi:10.1137/17M116152X

Journal Article Type Article
Acceptance Date Aug 23, 2018
Online Publication Date Oct 23, 2018
Publication Date Oct 23, 2018
Deposit Date Aug 24, 2018
Publicly Available Date Aug 24, 2018
Electronic ISSN 1936-4954
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 11
Issue 4
Pages 2368-2428
DOI https://doi.org/10.1137/17M116152X
Public URL https://nottingham-repository.worktribe.com/output/1047226
Publisher URL https://epubs.siam.org/doi/10.1137/17M116152X

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